On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts

We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_(y_1, y_2, C) > 0 such that for all T >= T_, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0. This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron’s formula.